To find the perimeters of the two triangles, we start with the first triangle which has side lengths of 6, 6, and 9 units. The perimeter \( P_1 \) of this triangle can be calculated as follows:
\[ P_1 = 6 + 6 + 9 = 21 \text{ units} \]
Next, we calculate the perimeter \( P_2 \) of the larger triangle with side lengths of 12, 12, and 18 units:
\[ P_2 = 12 + 12 + 18 = 42 \text{ units} \]
Now, we need to find the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle:
\[ \text{Ratio} = \frac{P_1}{P_2} = \frac{21}{42} \]
This ratio simplifies to:
\[ \frac{21}{42} = \frac{1}{2} \]
Therefore, the perimeter of the smaller triangle is half that of the larger triangle, and the final answer is:
\[ \text{The ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle is } \frac{1}{2}. \]
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